# What is the general solution of the differential equation  xyy'=x^2+1  ?

Mar 23, 2017

${y}^{2} = {x}^{2} + 2 \ln | x | + A$

#### Explanation:

The differential equation

$x y y ' = {x}^{2} + 1$

is a First Order linear separable Differential Equation which can be solved simply by rearranging and collection term in $x$ on the RHS and term in $y$ on the LHS;

$y \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{x}^{2} + 1}{x}$

And now we "separate the variables" to get;

$\int \setminus y \setminus \mathrm{dy} = \int \setminus x + \frac{1}{x} \setminus \mathrm{dx}$

Which is trivial to integrate to get:

$\setminus \setminus \frac{1}{2} {y}^{2} = \frac{1}{2} {x}^{2} + \ln | x | + C$

$\therefore {y}^{2} = {x}^{2} + 2 \ln | x | + 2 C$
$\therefore {y}^{2} = {x}^{2} + 2 \ln | x | + A$